Verbs kind of confuse me so excuse me if this sounds like a dumb question but:

She allowed him one meal a day

vs.

She let him one meal a day

Why exactly do I have to include the verb have in the second sentence?

Why is it that I can include the verb have in the first sentence as well but that it has to be in full infinitive form ("She allowed him have one meal a day" vs. "She allowed him to have one meal a day")?

It seems to me that allowed and let are not syntactically synonyms. Similar to watch and look. While similar concepts. The sentences "I watched TV." and "I looked at TV." are both correct, but we had to add the at.

Allowed seems like it implies a quantity already, where let is more like permitted. It's missing something, "She permitted him one meal a day." makes only slightly more sense than "She let him one meal a day.". Permitting and letting both don't specify what action is permitted, and so require another verb. Allowed gets away with this because it's like saying "let have" or "permitted to have" (while the sentence will sometimes need to be reorganized, like "permitted him to have"). The have concept is already included?